The Hilbert 16th problem was proposed as one of 23 famous problems in mathematics in 1900. It remains open until now. The Hilbert 16th problem has two parts: classification of the on ovals, which are defined by a polynomial equation \(\lbrace H(x, y) = 0\rbrace \) , and the number of limit cycles of polynomial vector fields. In this paper, we focus on the problems related to the
second part([1]}, [2]}). The most important topic of the second part is to find the upper bounded number of limit cycles, which is one of the main themes of the quantitative theory of ordinary differential equations (see, e.g., [3]}, [4]}, [5]}, [6]}, [1]}, [8]}, [9]}, [10]}, [2]}, [12]}, [13]}, [14]}, [15]}, [16]}, [17]}, [18]}, [19]}). Since the Hilbert 16th problem is notably difficult, it remains open (see [13]}),
Smale [15]} suggested to first solve the number of limit cycles of the polynomial Liénard systems.
In fact, the Liénard system is a notably common system in engineering and can exhibit notably rich dynamics. The investigation of Liénard systems has a long history and many results for the limit cycles (see [22]}). Rychkov [14]} studied a Liénard system as follows.
\(\left\lbrace \begin{array}{l}\displaystyle \frac{dx}{dt}=y-(\mu _1x+\mu _2x^3+x^5), \\\displaystyle \frac{dy}{dt}=-x,\end{array}\right.\) 
